About Binomial Distribution Calculator
This binomial distribution calculator counts successes in a fixed number of independent yes or no trials with the same chance of success. It opens on 10 fair coin flips and finds the chance of at most 3 heads. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
Use it when there are n trials, each with two outcomes, a constant success probability p and independent results. Quality checks, survey yes answers and free throws all fit.
Discrete tails need care. P(X < 3) stops at 2 successes while P(X ≤ 3) includes 3, and "at least 3" is P(X ≥ 3). The bar chart highlights exactly the counts that are included.
How to use Binomial Distribution Calculator
- Choose the distribution from the list. Each has its own parameters, such as the mean and standard deviation for the normal.
- Enter the parameters. The hint under each field says what is allowed.
- Choose what to calculate: a lower tail, an upper tail, an interval, a density or point probability, or the inverse.
- Enter x, or a and b for an interval, or the probability for the inverse. The result updates as you type.
- Read the answer, the mean and variance, and the shaded chart. Open the working to see which CDF values were used.
- Copy the result with the copy button.
Features
- Nine distributions: normal, binomial, Poisson, geometric, exponential, uniform, Student t, chi-square and F.
- Lower and upper tails, strict and non-strict, and intervals P(a ≤ X ≤ b).
- Density for continuous and point probability for discrete distributions.
- Inverse (quantile) calculation, including critical values for t, chi-square and F.
- Mean, variance and standard deviation, shown as undefined where they do not exist.
- Bell curve with shaded tail or interval for continuous distributions, and a bar chart for discrete ones.
- Upper tails computed directly, so very small p-values keep their precision.
- Formula of the distribution in KaTeX and a short worked step for every answer.
Binomial probability mass function
The chance of exactly k successes is the number of ways to place them times the probability of each arrangement.
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)
mean = n × p
variance = n × p × (1 − p)Worked examples
10 coin flips, p = 0.5: P(X ≤ 3) gives 0.17188
(1 + 10 + 45 + 120) ÷ 1024 = 0.171875.
20 products, 5% defective: P(X ≥ 2) gives 0.26416
The chance that a batch of 20 contains at least two defective items.
Privacy: how it works
- Everything is calculated by code running in your browser tab. Nothing you enter is sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- For a discrete distribution, P(X < 3) and P(X ≤ 3) are different. For a continuous one they are the same.
- For a two-sided critical value, such as the 95% value of t, put the inverse probability at 0.975.
- A density can be above 1 when the spread is small. It is not a probability.
- If your question is "at least k", use P(X ≥ x), not P(X > x).
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I calculate "at least" with a binomial?
Choose P(X ≥ x). For at least 2 defects in 20 items, enter x = 2. The calculator uses the upper tail directly.
When can I use the normal approximation?
When n × p and n × (1 - p) are both about 10 or more. You do not need it here, because the calculator gives the exact binomial value.
What are the conditions for a binomial distribution?
A fixed number of trials, two outcomes per trial, the same success probability each time, and independent trials.
More Probability Distribution Calculator pages
- Probability Distribution CalculatorPick a distribution, set its parameters and get PDF, CDF, interval probabilities, quantiles, mean and variance with a chart.
- Normal Distribution CalculatorEnter a mean and standard deviation to get tail probabilities, intervals and percentiles on a bell curve.
- Poisson Distribution CalculatorEnter the average rate λ to get the chance of exactly k events, at most k, at least k or a range.
- Geometric Distribution CalculatorEnter the success probability to get the chance that the first success happens on or by trial k.
- Exponential Distribution CalculatorEnter the rate λ to get the chance a waiting time exceeds, or stays below, a value, and its percentiles.
- Uniform Distribution CalculatorEnter the minimum a and maximum b to get the chance of landing in any sub-interval.
- Student t Distribution CalculatorEnter the degrees of freedom to get t critical values, tail areas and p-values on a shaded t curve.
- Chi-Square Distribution CalculatorEnter the degrees of freedom to get chi-square p-values, critical values and tail areas.
- F Distribution CalculatorEnter both degrees of freedom to get F p-values, critical values and tail areas.